Database instances as Set-valued functors

Just like all sets are instances on the schema 1, all functions are instances on the schema 2.

Database instance(1)

A \(\mathcal{C}\) instance, where \(\mathcal{C}\) is a schema, i.e. a finitely-presented category.

A functor \(\mathcal{C} \xrightarrow{I} \mathbf{Set}\)

Linked by

Grph as functor category: referenced

Database instance example(1)

Consider the following category: \(\boxed{\overset{\bullet}{z}\circlearrowleft s\ \ \boxed{s;s = s}}\)
A functor from this category to Set is ?a set \(Z\) and ?a involution function \(Z \rightarrow Z\).
- \(Z =\) natural numbers and a function sending everything to zero (zero is sent to zero)
- \(Z =\) set of well-formed arithmetic expressions (e.g. \(12+(2*4)\)) and a function which evaluates to an integer (which itself is a well-formed expression). Evaluation on integers does nothing.
- A function which sends numbers greater than 2 to their smallest prime factor (this is a no-op on the prime factors themselves).

Exercise 3-45(2)

For any functor \(\mathbf{1} \xrightarrow{F} \mathbf{Set}\) one can extract a set, \(F(1)\). Show that for any set \(S\) there is a functor \(\mathbf{1}\xrightarrow{F_S}\mathbf{Set}\) such that \(F_S(1)=S\)

Solution(1)

Define \(F_S\) to send the object of 1 to \(S\) and preserve identity morphisms. There is no nontrivial composition to enforce, so this is a valid functor.